Fitting Sets in U-Groups Olivier Frécon

Home , Ascendant subgroup, Subnormal subgroup

Fitting sets in U-groups Olivier Frécon

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Olivier Frécon. Fitting sets in U-groups. 2013. hal-00912367

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FITTING SETS IN U-GROUPS

OLIVIER FRECON´

Abstract. We consider different natural definitions for Fitting sets, and we prove the existence and conjugacy of the associated injectors in the largest possible classes of locally finite groups.

A Fitting set X of a finite group G is a non-empty set of subgroups which is closed under taking normal products, taking subnormal subgroups and conjugation in G. This concept of Anderson generalized the work of Fischer, Gaschütz and Hartley about Fitting classes [2, p.535–537], where a Fitting class is a class of finite groups closed under taking normal products and taking subnormal subgroups. The theory of Fitting sets concern the study of injectors: for a collection X of finite groups, a subgroup V of a finite group G is an X-injector of G if V ∩ A is a maximal X-subgroup of A for every subnormal subgroup A. The basic theorem provides the existence and conjugacy of X-injectors in any finite soluble group G for each Fitting set X [2, Theorem 2.9 p.539]. In this paper, we consider the natural generalizations of Fitting sets and injectors to locally finite groups. For each definition of a Fitting set, we prove the existence and conjugacy of associated injectors in a class of locally finite groups. It is shown in [3] that our classes of locally finite groups are the largest in which the results about injectors in finite soluble groups hold. Note that, thanks to our new approach, we generalize in the last section the theorem of Hartley and Tomkinson about the existence and conjugacy of injectors in U-groups [6] from Fitting classes to Fitting sets.

1. Normal Fitting sets In this section, we analyze the injectors associated to normal Fitting sets. Notation 1.1. Let X be a collection of groups. For each group G, we denote by GXn the join of its normal X-subgroups. Definition 1.2. Let G be a locally finite group. A nonempty set X of subgroups of G is a normal Fitting set of G if: (NF 1) every normal subgroup of an X-group belongs to X; (NF 2) when H is a subgroup of G then HXn is an X-subgroup of H; (NF 3) every conjugate of an X-group is an X-group. Definition 1.3. Let G be a locally finite group. If X is a set of subgroups of G, a subgroup V of G is an X-n-injector of G if, for every normal subgroup A of G, V ∩ A is a maximal X-subgroup of A.

Date: November 10, 2013. 1991 Mathematics Subject Classiﬁcation. Primary 20F17; Secondary 20D10. Key words and phrases. locally ﬁnite groups, Fitting sets, Injectors. 1 2 OLIVIER FRECON´

The main result of this section is the following. Theorem 1.4. Let G be a subsoluble U-group, and X a normal Fitting set of G. Then G has exactly one conjugacy class of X-n-injectors. It is likely that the class of subsoluble U-groups is the largest class of groups satisfying this result. A partial answer is obtained in [3]. We recall that a group is called a Baer group (resp. a Gruenberg group) if every cyclic subgroup is subnormal (resp. ascendant). A group G is said to be subsoluble (resp. an SN ∗-group) if it possesses an ascending series with each term subnormal (resp. ascendant) in G and each factor abelian. In any group G, there is a Baer radical β(G) which is a Baer subgroup containing all the subnormal Baer subgroups of G, and a Gruenberg radical γ(G) which is a Gruenberg subgroup containing all the ascendant Gruenberg subgroups of G. We refer to [7, §2.3] for more details. As in [1, p.15], we denote by ρ(G) the Hirsch-Plotkin radical of any group G, and ρ0(G) = 1 and ρn+1(G)/ρn(G)= ρ(G/ρn(G)) for each integer n. We recall that the class U was introduced as below in [4], and that by an Hartley’s Theorem [1, Theorem 4.4.7 p.163], the first condition is redundant. Definition 1.5. The class U is the largest subgroup-closed class of locally finite groups satisfying the conditions: (U1) if G ∈ U then G = ρn(G) for an integer n; (U2) if G ∈ U and π is any set of primes, then the Sylow π-subgroups of G are conjugate in G. Fact 1.6. [5, Theorem E] If G is a U-group, then G/ρ(G) is (2-step soluble)-by-finite. In particular G/ρ(G) is soluble. Moreover, G/ρ(G) is hyperfinite. We need to introduce normal∗ Fitting sets as the sets X of subgroups of a locally finite group G satisfying the conditions (NF 1), (NF 3) and ∗ (NF 2 ) if H is a subgroup of G such that H/HXn is locally nilpotent, then HXn is a maximal X-subgroup of H. Any normal∗ Fitting set of a locally finite group G is a normal Fitting set too. By the result below, the converse is true in subsoluble U-groups. Proposition 1.7. If X is a normal Fitting set of a subsoluble U-group G, then X is a normal∗ Fitting set.

Proof. Let H be a subgroup of G such that H/HXn is locally nilpotent. We define δ0(H) = HXn, δi+1(H)/δi(H) = β(H/δi(H)) for every ordinal i, and δµ(H) = ∪i<µδi(H) for every limit ordinal µ. Since G is subsoluble, then H is subsoluble too, and there is an ordinal α such that δα(H)= H. Suppose toward a contradiction that HXn is not a maximal X-subgroup of H. Since it is an X-subgroup by (NF 2), there is a smallest ordinal γ such that HXn is strictly contained in an X-subgroup K of δγ (H). By (NF 1), we have K ∩ δi(H) = HXn for all i<γ, so there is an ordinal ν such that γ = ν + 1. Now K/HXn ≃ Kδν (H)/δν (H) is a Baer group. Hence, if g is an element of K, then hgiHXn is subnormal in K, and hgiHXn is an X-group by (NF 1). By the minimality of γ, it is a maximal X-subgroup of hgiδi(H) for all i<γ. For any element b of δν (H), the local nilpotence of H/HXn implies the nilpotence of hg,biHXn/HXn. Therefore hgiHXn is a subnormal X-subgroup of hg,biHXn. By FITTING SETS IN U-GROUPS 3 the maximality of hgiHXn in hgiδν (H), we have hgiHXn = (hg,biHXn)Xn and b normalizes hgiHXn, so hgiHXn =(hgiδν (H))Xn. Since δγ (H)/δν (H) is a Baer group, then hgiδν (H) is subnormal in δγ (H). Hence hgiHXn is a subnormal X-subgroup of δγ (H) and we have hgiHXn ≤ (δγ (H))Xn. As δγ (H) is a normal subgroup of H, we obtain hgiHXn ≤ HXn, contradicting the choice of g. This proves the proposition. For this section, we fix a normal∗ Fitting set F of a fixed U-group G. Lemma 1.8. Let A a subgroup of G containing G′. If A possesses a maximal F- subgroup W , then every F-subgroup of G containing W is contained in a maximal F-subgroup of G.

Proof. Let (Vi)i<α be an increasing sequence of F-subgroups of G containing W , for an ordinal α. For each i<α, since A is normal in G, Vi ∩ A is an F-subgroup of A containing W and we have Vi ∩ A = W by the maximality of W . As A contains ′ ′ G , then W contains V where V = ∪i<αVi. So Vi is normal in V for all i<α. We ∗ obtain V = VFn and V is an F-group by (NF 2 ). A Carter subgroup of a group H is a locally nilpotent and self-serializing subgroup of H. The main result concerning these subgroups is the following: Fact 1.9. [4, Theorem 5.4] Let H be a U-group, and A a normal subgroup of H. Then H has a unique conjugacy class of Carter subgroups. Moreover, if C is a Carter subgroup of H, then CA/A is a Carter subgroup of H/A and each Carter subgroup of H/A has this form. Lemma 1.10. Let A be a subgroup of G containing G′. If A has a maximal F- subgroup W , then the maximal F-subgroups of G containing W are conjugate. Proof. By Lemma 1.8, G has a maximal F-subgroup V containing W , and W = V ∩ A is normal in V . Let U = NG(W ) and N/W = NU/W (V/W ). We show that there exists a Carter subgroup C/W of U/W such that V = CFn. As [V,N] is contained in V ∩ G′ ≤ W , N/W centralizes V/W . By Fact 1.9, N/W has a Carter subgroup C/W . Then C/W contains V/W , and V is a normal and maximal F-subgroup of C. In particular, we obtain V = CFn and V/W is normal in NU/W (C/W ). Thus we have NU/W (C/W ) = NN/W (C/W ) = C/W and C/W is a Carter subgroup of U/W . Let V1 and V2 be two maximal F-subgroups of G containing W . Then there are two Carter subgroups C1/W and C2/W of U/W such that V1 = (C1)Fn and V2 = (C2)Fn respectively. The conjugacy of C1 and C2 is given by Fact 1.9, and the lemma is proved. For every group H, we denote by H LN the intersection of all the normal subgroups K of H such that H/K is locally nilpotent. We recall that a subgroup H of a group G is pronormal in G if H and Hg are conjugate in hH,Hgi for every g ∈ G. Lemma 1.11. Let V be a pronormal subgroup of G such that V ∩GLN is normal in G. Then V/(V ∩ GLN) is a normal subgroup of a Carter subgroup of G/(V ∩ GLN).

LN Proof. We may assume V ∩ G = 1. Let G0 = G and, for all i ∈ N, Gi+1 = LN N Gi . We show that, for all i ∈ , VGi/Gi is a normal subgroup of a Carter subgroup of G/Gi. It is true if i = 0. Assume that the result is true for i ∈ N. 4 OLIVIER FRECON´

Then VGi/Gi is a normal subgroup of a Carter subgroup C/Gi of G/Gi. By the x c pronormality of V in G, for every c ∈ C, there is x ∈ Gi such that V = V , and we obtain C = NC (V )Gi. By Fact 1.9, NC (V ) has a Carter subgroup D0, and as C/Gi is locally nilpotent, we have C = D0Gi. Let D/Gi+1 be a maximal locally nilpotent subgroup of C/Gi+1 containing D0Gi+1/Gi+1. Now D/Gi+1 is a Carter subgroup of C/Gi+1 [4, Theorem 5.12], and D/Gi+1 is a Carter subgroup of LN LN G/Gi+1 by Fact 1.9. But we have (D0V ) ≤ V ∩ G = 1, hence D0V is locally nilpotent, and D0 contains V by the deﬁnition of a Carter subgroup. Therefore, VGi+1/Gi+1 is contained in a Carter subgroup D/Gi+1 of G/Gi+1. Since D/Gi+1 is locally nilpotent, VGi+1/Gi+1 is a serial subgroup of D/Gi+1. As VGi+1/Gi+1 is a pronormal subgroup of D/Gi+1, it is a normal subgroup of D/Gi+1. This proves that VGi+1/Gi+1 is a normal subgroup of a Carter subgroup of G/Gi+1 for each i ∈ N. In particular, V is a normal subgroup of a Carter subgroup of G.

Lemma 1.12. Let A be a subgroup of G containing G′, L a normal subgroup of G contaning GLN and V a maximal F-subgroup of G. Suppose that, if B is a normal subgroup of A and if B1 is a subgroup of B containing V ∩ B, then V ∩ B is an F-n-injector of B1. Then V ∩ L is a maximal F-subgroup of L. Proof. We show that V is pronormal in G. For every g ∈ G, V and V g are two maximal F-subgroups of hV,V gi. By our hypothesis, for all k ∈ N, V ∩ A(k) is an −1 F-n-injector of hV,V gi∩A(k) and of hV g ,V i∩A(k). So V g ∩A(k) is an F-n-injector of hV,V gi∩ A(k). Since there exists r ∈ N such that A(r) is locally nilpotent (Fact g (r) g (r) ∗ 1.6), the unique F-n-injector of hV,V i∩ A is (hV,V i∩ A )Fn by (NF 2 ), and we obtain V ∩ A(r) = V g ∩ A(r). Suppose that for an element k ∈ {1,...,r} there exists α ∈ A(k) such that V ∩ A(k) = V gα ∩ A(k). By hypothesis, V ∩ A(k) is an F-n-injector of hV,V gαi∩ A(k). Then V ∩ A(k−1) and V gα ∩ A(k−1) are conjugate in A(k−1) by Lemma 1.10. Thus V ∩ A and V g ∩ A are conjugate in A. Now a last application of Lemma 1.10 provides the conjugacy of V and V g in hV,V gi, as desired. Let N = NG(V ∩ L ∩ A). Since V ∩ L ∩ A is an F-n-injector of L ∩ A, Lemma 1.8 shows that V ∩L is contained in a maximal F-subgroup V1 of L. We show that V1 is g pronormal in N. By hypothesis, V1 ∩A = V ∩L∩A is an F-n-injector of hV1,V1 i∩A g for all g ∈ G. So, for every g ∈ N, V1 and V1 are two maximal F-subgroups of g g g hV1,V1 i ≤ L containing the F-n-injector V1 ∩ A = V1 ∩ A of hV1,V1 i∩ A. By g g Lemma 1.10, we obtain conjugacy of V1 and V1 in hV1,V1 i. Hence V1 is pronormal in N. LN LN LN Since G is contained in A and L, V ∩ N and V1 ∩ N are contained in LN LN V ∩L∩A = V1 ∩A. So V ∩N = V1 ∩N is a normal subgroup of N. By Lemma LN LN 1.11, V/(V ∩ N ) and V1/(V ∩ N ) are normal subgroups of Carter subgroups LN LN LN C/(V ∩ N ) and C1/(V ∩ N ) of N/(V ∩ N ) respectively, and Fact 1.9 gives x x ∈ N such that C1 = C. As V is a maximal F-subgroup of G and as CFn ∈ F x x x contains V and V1 , we ﬁnd V1 ≤ V . But V1 is a maximal F-subgroup of L, so x V ∩ L = V1 is a maximal F-subgroup of L.

Lemma 1.13. Let A be a subgroup of G containing G′ and let V be a maximal F-subgroup of G. We suppose that, if B is a normal subgroup of G contained in A, then V ∩ B is an F-n-injector of B. Then V ∩ L is a maximal F-subgroup of L for every normal subgroup L of G such that G/L is soluble. FITTING SETS IN U-GROUPS 5

Proof. We proceed by induction on the solubility class k of G/L. We may assume k ≥ 1. Let M = G(k−1)L. By induction, V ∩ M is a maximal F-subgroup of M. Since V ∩ M ′ is an F-n-injector of M ′ by our hypothesis, Lemma 1.8 says that ′ V ∩ M is contained in a maximal F-subgroup V1 of L. Since V1 is contained in a maximal F-subgroup V2 of M (Lemma 1.8), we have two maximal F-subgroups ′ V ∩ M and V2 of M containing V ∩ M . Lemma 1.10 gives g ∈ M such that g g (V ∩ M) = V2. As L is normal in G, we obtain (V ∩ L) = V2 ∩ L = V1, and V ∩ L is a maximal F-subgroup of L.

The following proposition is an analogue of [2, Proposition 2.12].

Proposition 1.14. Let (Ai)i=0,...,n (n ∈ N) be a ﬁnite increasing series of subgroups of G such that A0 is locally nilpotent and normal, G/A0 is soluble, An = G and, for every i ∈ {0,...,n − 1}, Ai is normal in Ai+1 and Ai+1/Ai is abelian. Suppose that there is a subgroup V of G such that V ∩ A0 =(A0)Fn and such that V ∩ Ai is a maximal F-subgroup of Ai for every i ∈ {1,...,n}. If B is a normal subgroup of G and if B1 is a subgroup of B containing V ∩ B, then V ∩ B is an F-n-injector of B1. Proof. We proceed by induction on n. We may assume that G is not locally nilpotent and n ≥ 1. First, we show that V is an F-n-injector of G. Let L be a normal subgroup of G. By induction, for i ∈ {0, ··· ,n − 1}, if N is a normal subgroup of Ai, and if N1 is a subgroup of N containing V ∩N, then V ∩N is an F-n-injector of N1. In particular, V ∩ ρ(G)L ∩ Ai is a maximal F-subgroup of ρ(G)L ∩ Ai, and we have V ∩ ρ(G)L ∩ A0 = ρ(G)L ∩ (A0)Fn =(ρ(G)L ∩ A0)Fn. Moreover, by Lemma 1.13 and Fact 1.6, V ∩ ρ(G)L is a maximal F-subgroup of ρ(G)L. By induction, if E is a normal subgroup of ρ(G)L ∩ An−1 and if E1 is a subgroup of E containing V ∩ E, then V ∩ E is an F-n-injector of E1. By an application of Lemma 1.12 to ρ(G)L, V ∩ L is a maximal F-subgroup of L. Hence V is an F-n-injector of G. Let B be a normal subgroup of G and let B1 be a subgroup of B containing V ∩ B. We show that V ∩ B is an F-n-injector of B1. As V is an F-n-injector of G, V ∩ B ∩ Ai = V ∩ B1 ∩ Ai is a maximal F-subgroup of B ∩ Ai and of B1 ∩ Ai for all i ∈ {0,...,n}, and we have

V ∩ B1 ∩ A0 = V ∩ B ∩ A0 = B ∩ (A0)Fn =(B1 ∩ A0)Fn.

By induction, if D is a normal subgroup of B1 ∩ An−1 and if D1 is a subgroup of D containing V ∩ D, then V ∩ D is an F-n-injector of D1. In particular, if D is a normal subgroup of B1 contained in B1 ∩ An−1, then V ∩ D is an F-n-injector of D. Let L be a normal subgroup of B1. Then, by Lemma 1.13 and Fact 1.6, V ∩ ρ(B1)L ∩ Ai is a maximal F-subgroup of ρ(B1)L ∩ Ai for all i ∈ {0,...,n} and we have

V ∩ ρ(B1)L ∩ A0 = ρ(B1)L ∩ (A0)Fn =(ρ(B1)L ∩ A0)Fn.

By induction, if F is a normal subgroup of ρ(B1)L ∩ An−1 and if F1 is a subgroup of F containing V ∩ F , then V ∩ F is an F-n-injector of F1. By an application of Lemma 1.12 to ρ(B1)L, V ∩ L is a maximal F-subgroup of L. Hence V ∩ B is an F-n-injector of B1.

Corollary 1.15. If G possesses an F-n-injector V , then V ∩ A is a maximal F- subgroup of A for every subnormal subgroup A of G. 6 OLIVIER FRECON´

Proof. We may assume A normal in G. Then V ∩ A(k) is a maximal F-subgroup of A(k) for every k ∈ N. Let n ∈ N be such that A(n) is locally nilpotent (n exists (n) (n) ∗ by Fact 1.6). Then we have V ∩ A =(A )Fn by (NF 2 ), and Proposition 1.14 says that V ∩ A is an F-n-injector of A. Theorem 1.16. G possesses a unique conjugacy class of F-n-injectors. Proof. We proceed by induction on the solubility class k of G/ρ(G). By (NF 2∗) we may assume that G is not locally nilpotent, that is k ≥ 1. By induction, G′ has an F-n-injector W , and W is contained in a maximal F-subgroup V of G by Lemma 1.8. In particular, V ∩ G(i) = W ∩ G(i) is a maximal F-subgroup of G(i) for (k) (k) every i ∈ {0,...,k}, and we have V ∩ G = (G )Fn. Hence V is an F-n-injector of G by Proposition 1.14. ′ ′ Let V1 and V2 be two F-n-injectors of G. Then V1 ∩ G and V2 ∩ G are F-n- ′ ′ ′ injectors of G (Corollary 1.15), and we can suppose V1 ∩G = V2 ∩G by induction. Lemma 1.10 gives the result. Now we obtain Theorem 1.4 by Proposition 1.7 and Theorem 1.16. Remark 1.17. If the word normal is replaced by subnormal in Deﬁnition 1.2 then clearly we do not obtain a new notion of Fitting set. Moreover, Corollary 1.15 shows that if we do the same thing in Deﬁnition 1.3, then in the context of subsoluble U- groups, we do not obtain a new notion of injector.

2. Ascendant Fitting sets In this section we replace the word normal in §1 by ascendant.

Notation 2.1. Let X be a collection of groups. For each group G, we denote by GXa the join of its ascendantl X-subgroups. Definition 2.2. Let G be a locally finite group. A nonempty set X of subgroups of G is an ascendant Fitting set of G if it satisfies (NF 3) and: (AF 1) every ascendant subgroup of an X-group belongs to X; (AF 2) when H is a subgroup of G then HXa is an X-subgroup of H. Definition 2.3. Let G be a locally finite group. If X is a set of subgroups of G, a subgroup V of G is an X-a-injector of G if, for every ascendant subgroup A of G, V ∩ A is a maximal X-subgroup of A. We will prove the following result. Theorem 2.4. Let G be both an SN ∗-group and a U-group, and X an ascendant Fitting set. Then G has exactly one conjugacy class of X-a-injectors. It is likely that the class SN ∗ ∩ U is the largest class of groups satisfying this result. A partial answer is obtained in [3]. Similarly to §1, we define an ascendant∗ Fitting set as a set X of subgroups of a locally finite group G satisfying (AF 1), (NF 3) and ∗ (AF 2 ) if H is a subgroup of G such that H/HXa is locally nilpotent, then HXa is a maximal X-subgroup of H. We note that in any locally finite group G, an ascendant Fitting set is a normal Fitting set, and an ascendant∗ Fitting set is both a normal∗ Fitting set and an ascendant Fitting set. FITTING SETS IN U-GROUPS 7

Moreover, by a similar proof of Proposition 1.7, we show that if G is both an SN ∗-group and a U-group, then its an ascendant∗ Fitting sets are precisely its ascendant Fitting sets. Theorem 2.5. Let G be a U-group. For any ascendant∗ Fitting set F of G, the F-n-injectors of G are precisely its F-a-injectors.

Proof. Let A be an ascendant subgroup of G. Let (Ai)i≤α (α ordinal) be an ascending series from A to G. Let W be an F-n-injector of A (Theorem 1.16). We prove that there is an ascending series (Wi)i≤α such that W0 = W and Wi is an F-n-injector of Ai for every i ≤ α. Assume Wi has been constructed. Let Ui+1 be an F-n-injector of Ai+1. Then Ui+1 ∩ Ai is an F-n-injector of Ai by Corollary a a 1.15. By Theorem 1.16, there is a ∈ Ai such that (Ui+1 ∩ Ai) = Wi. So Ui+1 is a an F-n-injector of Ai+1 containing Wi, and we can take Wi+1 = Ui+1. We must show that, if j is a limit ordinal, then Wj = ∪i

3. Serial Fitting sets In this section we replace the word normal in §1 by ascendant.

Notation 3.1. Let X be a collection of groups. For each group G, we denote by GX the join of its serial X-subgroups. Definition 3.2. Let G be a locally finite group. A nonempty set X of subgroups of G is a serial Fitting set of G if it satisfies (NF 3) and: (SF 1) every serial subgroup of an X-group belongs to X; (SF 2) when H is a subgroup of G then HXs is an X-subgroup of H. We note that in any locally finite group G, a serial Fitting set is an ascendant∗ Fitting set too. Definition 3.3. Let G be a locally finite group. If X is a set of subgroups of G, a subgroup V of G is an X-injector of G if, for every serial subgroup A of G, V ∩ A is a maximal X-subgroup of A. We will prove the following result. 8 OLIVIER FRECON´

Theorem 3.4. Let G be a U-group, and X a serial Fitting set of G. Then G has exactly one conjugacy class of X-injectors. It is proven in [3] that the class of U-groups is the largest class of groups satisfying this result. The difficulty of the proof in [3] is that without the conjugacy of Sylow subgroups, a Sylow subgroup might not be an injector. From now on, G is a fixed U-group and F is a serial Fitting set of G. Lemma 3.5. Let N be a normal subgroup of G such that G/N is a p-group for a prime p and NF is a maximal F-subgroup of N. If N contains a subgroup A of G and if V is an F-a-injector of G, then V ∩ A is an F-a-injector of A. Proof. By Theorems 1.16 and 2.5, A has an F-a-injector W . Then we have W ∩N = NF, so W/NF is a p-group and there is a maximal p-subgroup R/NF of G/NF containing W/NF. Likewise we have V ∩N = NF and there is a maximal p-subgroup S/NF of G/NF containing V/NF. As G/N is a p-group, we have G = RN = SN g [4, Lemma 2.1 (ii)], and there is g ∈ N such that R = S. But S/NF is locally nilpotent, so SF contains all the F-subgroups of S containing NF. Hence SF contains g g g V and W . By the maximality of V , we obtain V = SF and W ≤ V . But W is an F-a-injector of A since g ∈ N ≤ A. As A contains N and as G/N is locally nilpotent, A is a serial subgroup of G. Thus V ∩ A is an F-subgroup of A. Since V ∩ A contains an F-a-injector W g of A, we obtain V ∩ A = W g and the proof is complete. Lemma 3.6. Assume that G has a normal p-subgroup M for a prime p and a serial subgroup A such that G = MA. If V is an F-a-injector of G, then V ∩ A is an F-a-injector of A. Proof. Let P = hS : S is a p′-subgroup of Gi. We show that P is a subgroup of A. By the seriality of A, for each x ∈ G \ A, there exist U and V two subgroups of G containing A such that V E U and x ∈ U \ V . But U = (M ∩ U)V , so U/V is a p-group and x is not a p′-element. This proves that each p′-subgroup of G is contained in A, hence P is a subgroup of A. We show that V ∩NA(V ∩P ) is an F-a-injector of NA(V ∩P ). Let G1 = NG(V ∩P ) and N = NP (V ∩P ). As P is normal in G, V ∩P is an F-a-injector of P , so we have NF = V ∩ P , and NF is a maximal F-subgroup of P and of N. Since N = G1 ∩ P ′ contains all the p -elements of G1, G1/N is a p-group. But, as A contains P , NA(V ∩ P ) is a subgroup of G1 containing N and, since V is an F-a-injector of G1 (Corollary 1.15 and Theorem 2.5), Lemma 3.5 says that V ∩ NA(V ∩ P ) is an F-a-injector of NA(V ∩ P ). By Theorems 1.16 and 2.5, A has an F-a-injector W . Then W ∩P and V ∩P are two F-a-injectors of P and there is a ∈ P such that (W ∩ P )a = V ∩ P (Theorem a 2.5). Now W is an F-a-injector of NA(V ∩ P ) by Corollary 1.15 and Theorem 2.5. Since V ∩ NA(V ∩ P ) is an F-a-injector of NA(V ∩ P ), Theorem 2.5 says that a W and V ∩ NA(V ∩ P ) are conjugate in NA(V ∩ P ). Thus V ∩ NA(V ∩ P ) is an F-a-injector of A contained in the F-subgroup V ∩A. Hence V ∩NA(V ∩P )= V ∩A and V ∩ A is as desired. Theorem 3.7. The F-a-injectors of G are precisely its F-injectors. Proof. It is sufficient to prove that any F-a-injector V of G is an F-injector. We prove that, if A is a serial subgroup of G, then V ∩ A is an F-a-injector of A. Let FITTING SETS IN U-GROUPS 9

W be an F-a-injector of A. We show that W is contained in an F-a-injector U of Aρ(G). Let N0 = 1 and, for every j ≥ 1, Nj = Oπj (ρ(G)) where πj = {q : th q is a the i prime for i ≤ j}. We construct an increasing sequence (Wi)i∈N of F-subgroups of G such that W0 = W and, for every i ∈ N, Wi is an F-a-injector of NiA. Suppose Wi constructed for i ∈ N. Let Wi+1,0 be an F-a-injector of Ni+1A. th Let p be the (i + 1) prime. Then we have Ni+1 = Ni ×Op(ρ(G)), and since NiA is a serial subgroup of Ni+1A, then Wi+1,0 ∩ NiA is an F-a-injector of NiA by u Lemma 3.6. So there is u ∈ NiA such that Wi =(Wi+1,0 ∩ NiA) (Theorem 2.5). u Now we may choose Wi+1 = Wi+1,0, and Wi = Wi+1 ∩ NiA is serial in Wi+1. Let U = ∪i∈NWi. Then Wi is a serial F-subgroup of U for each i, and U = UF is an F-group. Moreover we have U ∩ NiA = Wi for every i ∈ N. Let B be an ascendant subgroup of Aρ(G) and U1 be an F-subgroup of B containing U ∩ B. For every i ∈ N, as NiA is a serial subgroup of Aρ(G), U1 ∩ NiA is an F-group containing Wi ∩ B. But Wi ∩ B is a maximal F-subgroup of NiA ∩ B for every i ∈ N since Wi is an F-a-injector of NiA. So we have U1 ∩ NiA = Wi ∩ B for every i ∈ N, and U1 = ∪i∈N(U1 ∩ NiA) = ∪i∈N(Wi ∩ B) = U ∩ B. Hence U is an F-a-injector of Aρ(G) containing W . It follows from Fact 1.6 that any serial subgroup of G/ρ(G) is acendant [1, Lemma 7.2.11], so Aρ(G) is ascendant in G, and V ∩ Aρ(G) is an F-a-injector of Aρ(G). By Theorem 2.5, there is g ∈ Aρ(G) such that U g = V ∩ Aρ(G). But N g g there is k ∈ such that g ∈ NkA and we have V ∩ NkA =(U ∩ NkA) = Wk . So V ∩ NkA is an F-a-injector of NkA. Now, by successive applications of Lemma 3.6, V ∩ A is an F-a-injector of A. Now we obtain Theorem 3.4 from Theorems 1.16, 2.5 and 3.7. References

[1] Martyn R. Dixon. Sylow theory, formations and Fitting classes in locally finite groups, volume 2 of Series in Algebra. World Scientific Publishing Co. Inc., River Edge, NJ, 1994. [2] Klaus Doerk and Trevor Hawkes. Finite soluble groups, volume 4 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1992. [3] Olivier Frécon. On the conjugacy of injectors in locally finite groups. Preprint. [4] A. D. Gardiner, B. Hartley, and M. J. Tomkinson. Saturated formations and Sylow structure in locally finite groups. J. Algebra, 17:177–211, 1971. [5] B. Hartley. Sylow subgroups of locally finite groups. Proc. London Math. Soc. (3), 23:159–192, 1971. [6] B. Hartley and M. J. Tomkinson. Carter subgroups and injectors in a class of locally finite groups. Rend. Sem. Mat. Univ. Padova, 79:203–212, 1988. [7] Derek J. S. Robinson. Finiteness conditions and generalized soluble groups. Part 1. Springer- Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 62.

Laboratoire de Mathematiques´ et Applications,, UMR 7348 du CNRS,, Universite´ de Poitiers,, Tel´ eport´ 2 - BP 30179,, Bd Marie et Pierre Curie,, 86962 Futuroscope Chasseneuil Cedex,, France E-mail address: [email protected]